In contrast, in \(\mathcal {J}(\mathcal {A})\)  there is no canonical identification between self-adjoint operators and generators of Jordan automorphisms [1]}, [2]}. Consequently, in \(\mathcal {J}(\mathcal {A})\)  we cannot interpret the sign of the parameter \(t\)  in the corresponding one-parameter groups independently of the choice of Hamiltonian \(a \in \mathcal {J}(\mathcal {A})_\mathrm {sa}\) . By comparison, lifting \(\mathcal {J}(\mathcal {A})\)  to \(\mathcal {A}\)  thus equips the latter with an intrinsic direction of time, mediated by the identification \(a \mapsto \mathrm {ad}(ia)\) .Generalising the mapping \(a \mapsto \mathrm {ad}(ia)\) , [3]} characterise those maps which lift JB(W) to \(C^*\)  (von Neumann) algebras. By their physical interpretation, such maps are called dynamical correspondences. To emphasise this distinction, we define the canonical time orientation \(\Psi _\mathcal {A}\) on \(\mathcal {J}(\mathcal {A})\)  byThe notion of time orientation was introduced in [4]}, [5]}.
\(\Psi _\mathcal {A}:= \mathrm {Ad}: \mathbb {R}\times \mathcal {J}(\mathcal {A})_\mathrm {sa} \ni (t,a) \mapsto e^{t\mathrm {ad}(ia)},\) 
